扰动点理想格罗布纳基的近似，第一部分
An approximation of the Gröbner basis of ideals of perturbed points, part I
AI总结：
针对坐标精度有限的有限点集，提出一种预处理结合数值Buchberger-Möller算法的方法，近似计算多项式理想的Gröbner基，并保证近似基与精确基在首项上一致。
摘要：
我们开发了一种方法，用于近似计算在有限点集处消失的多项式理想的Gröbner基，其中点的坐标已知但精度有限。该方法包括对输入点的预处理阶段以减轻输入数据不确定性的影响，以及一种新的“数值”版本的Buchberger-Möller算法，用于计算精确Gröbner基的近似$ar{GB}$。第二部分基于一个阈值依赖的过程，从数值角度分析扰动向量属于扰动子空间的问题。通过适当选择阈值，集合$ar{GB}$被证明是“可能的”精确Gröbner基或精确基的“吸引子”基的良好近似。此外，$ar{GB}$的多项式与Stetter引入的扩展基的多项式“足够接近”，但它们的优势在于$LT(ar{GB})$与“可能的”精确情况的首项一致。预处理后的点集（未知精确点的近似）是$ar{GB}$多项式的伪零点集。
英文摘要：
We develop a method for approximating the Gröbner basis of the ideal of polynomials which vanish at a finite set of points, when the coordinates of the points are known with only limited precision. The method consists of a preprocessing phase of the input points to mitigate the effects of the input data uncertainty, and of a new "numerical" version of the Buchberger-Möller algorithm to compute an approximation $\bar{GB}$ to the exact Gröbner basis. This second part is based on a threshold-dependent procedure for analyzing from a numerical point of view the membership of a perturbed vector to a perturbed subspace. With a suitable choice of the threshold, the set $\bar{GB}$ turns out to be a good approximation to a "possible" exact Gröbner basis or to a basis which is an "attractor" of the exact one. In addition, the polynomials of $\bar{GB}$ are "sufficiently near" to the polynomials of the extended basis, introduced by Stetter, but they present the advantage that $LT(\bar{GB})$ coincides with the leading terms of a "possible" exact case. The set of the preprocessed points, approximation to the unknown exact points, is a pseudozero set for the polynomials of $\bar{GB}$.
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